Discretization.h

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// Copyright (c) FIRST and other WPILib contributors.
// Open Source Software; you can modify and/or share it under the terms of
// the WPILib BSD license file in the root directory of this project.
 
#pragma once
 
#include <unsupported/Eigen/MatrixFunctions>
 
#include "frc/EigenCore.h"
#include "units/time.h"
 
namespace frc {
 
/**
 * Discretizes the given continuous A matrix.
 *
 * @tparam States Number of states.
 * @param contA Continuous system matrix.
 * @param dt    Discretization timestep.
 * @param discA Storage for discrete system matrix.
 */
template <int States>
void DiscretizeA(const Matrixd<States, States>& contA, units::second_t dt,
                 Matrixd<States, States>* discA) {
  // A_d = eᴬᵀ
  *discA = (contA * dt.value()).exp();
}
 
/**
 * Discretizes the given continuous A and B matrices.
 *
 * @tparam States Number of states.
 * @tparam Inputs Number of inputs.
 * @param contA Continuous system matrix.
 * @param contB Continuous input matrix.
 * @param dt    Discretization timestep.
 * @param discA Storage for discrete system matrix.
 * @param discB Storage for discrete input matrix.
 */
template <int States, int Inputs>
void DiscretizeAB(const Matrixd<States, States>& contA,
                  const Matrixd<States, Inputs>& contB, units::second_t dt,
                  Matrixd<States, States>* discA,
                  Matrixd<States, Inputs>* discB) {
  // M = [A  B]
  //     [0  0]
  Matrixd<States + Inputs, States + Inputs> M;
  M.template block<States, States>(0, 0) = contA;
  M.template block<States, Inputs>(0, States) = contB;
  M.template block<Inputs, States + Inputs>(States, 0).setZero();
 
  // ϕ = eᴹᵀ = [A_d  B_d]
  //           [ 0    I ]
  Matrixd<States + Inputs, States + Inputs> phi = (M * dt.value()).exp();
 
  *discA = phi.template block<States, States>(0, 0);
  *discB = phi.template block<States, Inputs>(0, States);
}
 
/**
 * Discretizes the given continuous A and Q matrices.
 *
 * @tparam States Number of states.
 * @param contA Continuous system matrix.
 * @param contQ Continuous process noise covariance matrix.
 * @param dt    Discretization timestep.
 * @param discA Storage for discrete system matrix.
 * @param discQ Storage for discrete process noise covariance matrix.
 */
template <int States>
void DiscretizeAQ(const Matrixd<States, States>& contA,
                  const Matrixd<States, States>& contQ, units::second_t dt,
                  Matrixd<States, States>* discA,
                  Matrixd<States, States>* discQ) {
  // Make continuous Q symmetric if it isn't already
  Matrixd<States, States> Q = (contQ + contQ.transpose()) / 2.0;
 
  // M = [−A  Q ]
  //     [ 0  Aᵀ]
  Matrixd<2 * States, 2 * States> M;
  M.template block<States, States>(0, 0) = -contA;
  M.template block<States, States>(0, States) = Q;
  M.template block<States, States>(States, 0).setZero();
  M.template block<States, States>(States, States) = contA.transpose();
 
  // ϕ = eᴹᵀ = [−A_d  A_d⁻¹Q_d]
  //           [ 0      A_dᵀ  ]
  Matrixd<2 * States, 2 * States> phi = (M * dt.value()).exp();
 
  // ϕ₁₂ = A_d⁻¹Q_d
  Matrixd<States, States> phi12 = phi.block(0, States, States, States);
 
  // ϕ₂₂ = A_dᵀ
  Matrixd<States, States> phi22 = phi.block(States, States, States, States);
 
  *discA = phi22.transpose();
 
  Q = *discA * phi12;
 
  // Make discrete Q symmetric if it isn't already
  *discQ = (Q + Q.transpose()) / 2.0;
}
 
/**
 * Returns a discretized version of the provided continuous measurement noise
 * covariance matrix.
 *
 * @tparam Outputs Number of outputs.
 * @param R  Continuous measurement noise covariance matrix.
 * @param dt Discretization timestep.
 */
template <int Outputs>
Matrixd<Outputs, Outputs> DiscretizeR(const Matrixd<Outputs, Outputs>& R,
                                      units::second_t dt) {
  // R_d = 1/T R
  return R / dt.value();
}
 
}  // namespace frc